Finite-action solutions of Yang-Mills equations on de Sitter dS$_4$ and anti-de Sitter AdS$_4$ spaces

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter dS$_4$ and anti-de Sitter AdS$_4$ spaces and construct various solutions to the Yang-Mills equations. On de Sitter space we reduce the Yang-Mills equations via an SU(2)-equivariant ansatz to Newtonian mechanics of a particle moving in ${\mathbb R}^3$ under the influence of a quartic potential. Then we describe magnetic and electric-magnetic solutions, both Abelian and non-Abelian, all having finite energy and finite action. A similar reduction on anti-de Sitter space also yields Yang-Mills solutions with finite energy and action. We propose a lower bound for the action on both backgrounds. Employing another metric on AdS$_4$, the SU(2) Yang-Mills equations are reduced to an analytic continuation of the above particle mechanics from ${\mathbb R}^3$ to ${\mathbb R}^{2,1}$. We discuss analytical solutions to these equations, which produce infinite-action configurations. After a Euclidean continuation of dS$_4$ and AdS$_4$ we also present self-dual (instanton-type) Yang--Mills solutions on these backgrounds.


Introduction
Magnetic monopoles [1][2][3] and vortices [4] are playing an important role in the nonperturbative physics of 3+1 dimensional Yang-Mills-Higgs theory [5][6][7]. However, in pure gauge theory without any scalar fields there are no vortices or non-Abelian monopoles on Minkowski space R 3,1 . Yet, our universe appears to be asymptotically de Sitter (not Minkowski) at very early and very late times. This provides strong motivation for searching finite-action solutions in pure Yang-Mills theory on de Sitter space dS 4 . Finding Yang-Mills solutions on anti-de Sitter space AdS 4 is also reasonable from the viewpoint of string-theory applications and from the AdS/CFT perspective. The construction of such solutions, both Abelian and non-Abelian, is the goal of our paper. 1 Some steps in this direction have been made in [9].

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In this paper, we present a construction of smooth Abelian and non-Abelian solutions with both finite energy and action in pure Yang-Mills theory on de Sitter space dS 4 and anti-de Sitter space AdS 4 . Other types of Yang-Mills solutions on anti-de Sitter space, also described in this paper, have infinite energy and action. We also write down instantons and quasi-instantons in de Sitter dS 4 and anti-de Sitter AdS 4 spaces. We postpone the issue of boundary conditions and study classical solutions for any kind of boundary condition.
The paper is organized as follows. Section 2 provides a description of de Sitter space, which is used in section 3 to explicitly construct Yang-Mills solutions on dS 4 and to compute their energy and action. Instantons on the Euclideanized background are the topic of section 4. The story is repeated for anti-de Sitter space AdS 4 in sections 5, 6 and 7. We conclude in section 8. Four appendices list the various metrics used in the paper, detail metrics on the spatial slices S 3 and AdS 3 , and present explicit expressions for our Yang-Mills solutions in the fundamental and in the adjoint SU(2) representation on dS 4 in various coordinates.

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3 Yang-Mills configurations on dS 4 In Minkowski space R 3,1 smooth vortex or monopole solutions of gauge theory can be constructed only in the presence of Higgs fields. Both types of solutions have finite energy per length or energy, defined via integrals over R 2,0 ⊂ R 3,1 or R 3,0 ⊂ R 3,1 , respectively. There are no finite-energy solutions of such kind in pure Yang-Mills theory in Minkowski space. Here, we will show that finite-energy solutions in gauge theory without scalar fields do exist on de Sitter space dS 4 . Furthermore, they also have finite action, contrary to monopoles or vortices in R 3,1 .
Conformal invariance. Since in four dimensions the Yang-Mills equations are conformally invariant, their solutions on de Sitter space can be obtained by solving the equations on I × S 3 with the cylindrical metric (2.9). Therefore, we will consider a rank-N Hermitian vector bundle over the cylinder I × S 3 with a gauge potential A and the gauge field F = dA + A ∧ A taking values in the Lie algebra su(N ). The conformal boundary of dS 4 consists of the two 3-spheres at t = ± π 2 or, equivalently, at τ = ±∞. On a manifold M with boundary ∂M , gauge transformations are naturally restricted to tend to the identity when approaching ∂M (see e.g. [11]). This corresponds to a framing of the gauge bundle over the boundary. For our case, it means allowing only gauge-group elements g(·) subject to (3.1) Reduction to matrix equations. In order to obtain explicit solutions we use the SU(2)equivariant ansatz (cf. [12][13][14]) for the su(N )-valued gauge potential A in the temporal gauge A 0 ≡ A t = 0 = A τ . Here, X a (t) are three su(N )-valued functions depending only on t ∈ I, and e a are the basis one-forms on S 3 satisfying (2.4). The corresponding gauge field reads whereẊ a := dX a /dt and e 0 := dt. It is not difficult to show (see e.g. [14]) that the Yang-Mills equations on I × S 3 after substituting (3.2) and (3.3) reduce to the ordinary matrix differential equations Reduction to a Newtonian particle on R 3 . It is very natural to restrict the matrices X a to an su(2) subalgebra. To this end, we embed the spin-j representation of SU(2) into the fundamental of SU(N ) with N = 2j+1. The three SU(2) generators I a then obey [I b , I c ] = 2 ε a bc I a and tr(I a I b ) = −4 C(j) δ ab for C(j) =

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where C(j) is the second-order Dynkin index of the spin-j representation. The simplest choice for X a then is 3 where Ψ a are real functions of t ∈ I. Due to the equivalence of Yang-Mills theory on dS 4 with metric (2.8) to the theory on I × S 3 with metric (2.9), we obtain the Lagrangian density (3.7) The S 3 has disappeared, and we are left with at a Lagrangian on I. Interpreting the real functions Ψ a (t) as coordinates of a particle on R 3 , this Lagrangian describes its Newtonian dynamics in a finite time interval, with kinetic energy T and quartic potential energy V , 4 The critical points (Ψ 1 ,Ψ 2 ,Ψ 3 ) of this potential are where the number of minus signs in each triple must be even. The central minimum is isotropic with oscillation frequency ω = 2. The other four minima support eigenoscillations with frequencies ω = 2 and ω ⊥ = 4 with respect to the radial direction. The equations of motion can be obtained either by substituting (3.6) into (3.4) or from (3.7) as the Euler-Lagrange equations, (3.11) These equations are still difficult to solve. However, as can be seen from the contour plot of the potential in figure 1, the system enjoys tetrahedral symmetry. The permutation group S 4 acts on the triple (Ψ 1 , Ψ 2 , Ψ 3 ) ∈ R 3 by permuting the entries and by changing the JHEP11(2017)017 sign of an even number of entries. One may hope to find analytic solutions for trajectories fixed under part of this symmetry. The maximal subgroups of S 4 are A 4 (of order 2), D 8 (of order 3) and S 3 (of order 4). While A 4 leaves only the origin invariant, D 8 keeps fixed a coordinate axis (up to sign), and S 3 leaves invariant the direction to a noncentral potential minimum. Therefore, we look at two special cases. In the D 8 case, we pick the Ψ 3 -axis and consider Ψ 1 = Ψ 2 = 0 and Ψ 3 = ξ , (3.12) where ξ(t) is some real-valued function of t ∈ (− π 2 , π 2 ). With this simplification, we get showing that in this direction in parameter space the harmonic approximation is exact.
Here, the non-stable S 4 transformations act by permuting the coordinate axes, but two equivalent choices give the same equations. In the S 3 case, we choose the direction (1, 1, 1) and put where ψ(t) is some other real-valued function of t ∈ (− π 2 , π 2 ). This ansatz leads to the simplifications

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The remaining S 4 transformations flip the sign of two coordinates, which generates three other but equivalent configurations, yielding the same equations. Other directions fixed under some subgroup of S 4 do not give rise to elementary solutions.
Solutions. The general solution of the linear equation (3.13) is where γ and t 0 are arbitrary real parameters. Since only one su(2) generator is excited, it leads to an Abelian field configuration. We note that the normalization of the linear solution ξ(t) is arbitrary. For the non-Abelian ansatz (3.14) the simplest solutions of (3.15) are constant at the critical points of V ψ , i.e.
A prominent nontrivial solution of (3.15) is the bounce, which describes the motion from the local maximum (ψ=0) at t=−∞ on the cylinder R × S 3 to the turning point (ψ= √ 2) at t=t 0 and back to (ψ=0) at t=∞. Flipping the sign of ψ(t) produces the anti-bounce, which explores the other half of the double-well potential. In addition, there is a continuum of periodic solutions oscillating either about ψ = ±1 or exploring both wells of the double-well potential V ψ , which are given by Jacobi elliptic functions. Usually, the moduli parameter t 0 is trivial because of time translation invariance in (3.15). However, since for de Sitter space according to (2.7) we consider the solutions ψ(t) only on the interval I = (− π 2 , π 2 ) without imposing boundary conditions, the value of t 0 ∈ R makes a difference. It allows us to pick a segment of length π anywhere on the profile of the bounce, not necessarily including its minimum.
Explicit form of the Yang-Mills fields. Let us display the explicit non-Abelian Yang-Mills solutions on dS 4 corresponding to (3.17)-(3.18) after substituting (3.6) and (3.14) into (3.2) and (3.3). For ψ = ±1 we obtain the trivial solutions F ≡ 0 (vacua). For ψ = 0 we get the nontrivial smooth configuration with respect to the orthonormal basis (3.20), where we used (2.7). In appendix D we display the explicit coordinate dependence of these field components in different coordinates of S 3 .
Inserting the bounce solution (3.18), we obtain a family of nonsingular Yang-Mills configurations, depending on t 0 ∈ R. This family carries electric as well as magnetic fields. Their relative size as a function of τ is displayed in figure 2. Finally, for the Abelian solutions, the substitution of (3.16) yields

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Energy of the Yang-Mills solutions. The energy of Yang-Mills configurations on de Sitter space dS 4 computes as where S 3 (t) is the 3-sphere of radius R cos t . On the configuration (3.22), this evaluates to after additional use of (2.7). Similarly, for the configuration (3.23) one gets the same result (3.27) as for the purely magnetic configuration. The total energy of the Abelian configuration (3.24) and (3.25) is where γ 2 is the moduli parameter. We see that, for all these gauge-field configurations, the energy decays exponentially for early and late times. Its finiteness is quite obvious, since our configurations are non-singular on the finite-volume spatial S 3 slices.
Action of the Yang-Mills solutions. In a similar fashion one can evaluate the action functional on the field configurations (3.19b), (3.23b) and (3.24b). Due to conformal invariance, the action functional can be calculated either in the de Sitter metric (2.8) or in the cylinder metric (2.9) on I × S 3 . We have and the relation between e µ andẽ µ is given in (3.20). For the purely magnetic configuration (3.19b) the action evaluates to One may restore the gauge coupling in the denominator. The action on the 'bounce' configuration (3.23) comes out as Its numerical value varies between 5.52 (for δ=0) and −46.51 (for δ → ±∞). Finally, the action functional on the Abelian solutions (3.24) vanish, since the integrals of the electric and magnetic energy densitiesρ e andρ m are finite and equal.
In summary, we have described a class of Abelian and non-Abelian gauge configurations solving the Yang-Mills equations on de Sitter space dS 4 . They are spatially homogeneous and decay for early and late times. Their energies and actions are all finite.

Instantons on de Sitter space dS 4
A useful tool to obtain information about the non-perturbative dynamics of gauge theories in flat space is instanton configurations. It is known that by Euclidean continuation the space dS 4 becomes a 4-sphere S 4 of radius R with the metric (4.2). Therefore, instantons in dS 4 are the standard S 4 instantons. Here we present them in a form adapted to the coordinates on S 4 and the gauge A ϕ = 0.
Four-sphere. The Euclidean form of the dS 4 metric in global coordinates can be obtained by substituting Then the metric (2.3) becomes the metric on S 4 of radius R, This is the standard form in terms of four angles. By the coordinate transformation it is related to the stereographic coordinates so that (4.5)

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Conformal equivalence of metrics on S 4 and R×S 3 . The metric (4.2) or (4.5) is conformally equivalent to the metric on the Euclidean cylinder, respectively.
Self-duality. The instanton equations on S 4 , are conformally invariant, and it is more convenient to consider them on the cylinder R×S 3 with the metric In the basis (e i ) = (e a , dT ) the SU(2)-invariant (spherically symmetric) connection A in the gauge A T = 0 = A ϕ and its curvature are given by [15] A = X a e a , F 4a = dX a dT and and (4.9) reduces to a form of the generalized Nahm equations given by With the same ansatz as previously, these turn into a coupled set of three ordinary first-order differential equations (the dot denotes the derivative with respect to T ), with the superpotential  which is depicted in figure 3. The corresponding Newtonian dynamics is given bÿ yielding the potential given in (3.8), but entering with the opposite sign. Its critical points coincide with the potential minima listed in (3.9), with values U(0, 0, 0) = 0 and U(1, 1, 1) = 1. The flow equations (4.14) also imply that Instantons. The static trajectories Ψ a = 0 and Ψ a = 1 (and its images under S 4 ) lead to the trivial vacuum solution F = 0. However, there exists an analytic BPS solution interpolating between the two kinds of critical points of U . It is captured again by the further simplification which leaves us with a single differential equation,

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Its simplest solution is the kink with integration constant (or collective coordinate) T 0 , which produces By using (4.8), one can rewrite it as which is exactly the BPST instanton extended from R 4 to S 4 [15]. This is easily seen from where theẽ i form an orthonormal basis of one-forms on S 4 . For R = Λ (T 0 = 0), F has the canonical form of BPST instanton on S 4 . The radius R of S 4 sets the scale for Λ, but we may tune T 0 in (4.22) such as to remove the dependence of Λ on R. The action of this configuration evaluates to S = 8π 2 independent of R. The anti-instanton is found by flipping the sign of T .
Remark. Of course, the Newton equation (4.16) has more solutions than the flow equations (4.14). For example, other bounded solutions for ψ oscillate anharmonically between ψ = −1 and ψ = 1. When viewed in the full parameter space R 3 ∋ (Ψ a ) however, almost all classical trajectories will run away to infinity, since the inverted potential −V does not have any local minimum. This is reflected in the value of the topological charge which differs from zero or infinity only if the trajectory Ψ a (T ) connects the two types of critical points, i.e. for an instanton or anti-instanton. Those two saturate the inequality S ≥ 4π 2 C(j) |q|.

Description of anti-de Sitter space AdS 4
AdS 3 -slicing coordinates. For the remainder of the paper we attempt to repeat the previous analysis for anti-de Sitter space AdS 4 . In analogy with the dS 4 case, where we used the fact that S 3 ∼ = SU(2) is a group manifold, for AdS 4 we may employ instead another group manifold,

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and embed AdS 4 into R 3,2 in such a way that the metric on AdS 4 will be conformally equivalent to the metric on a cylinder R× PSL(2, R), in order to follow our recipe for constructing Yang-Mills solutions. So, One can introduce global coordinates (z, t, ρ, φ) by setting The flat metric on R 3,2 induces a metric on AdS 4 , where dΩ 2 2,1 denotes the metric on the unit-radius AdS 3 ∼ = PSL(2, R). On this space we introduce an orthonormal basis {e α } , α = 0, 1, 2, of left-invariant one-forms which satisfy the equations are the structure constants of the group SL(2, R). Concretely, from (5.6) we have In terms of e α the AdS 3 metric has the form dΩ 2 2,1 = η αβ e α e β = −(e 0 ) 2 + (e 1 ) 2 + (e 2 ) 2 . (5.8) Explicit formulae for coordinates and one-forms on unit AdS 3 can be found in appendix C.
Reduction to particle mechanics. As in section 3, we take the matrices X α from a spin-j representation of SU(2) with generators (I 1 , I 2 , I 3 ) inside su(N ) with N = 2j + 1 and put X 0 = Ψ 0 I 3 , X 1 = Ψ 1 I 1 and X 2 = Ψ 2 I 2 , (6.5) where Ψ α are real functions of χ. Substituting (6.5) into (6.3), we obtain for a quasi-potential function consistent with (6.4). However, the interpretation of a Newtonian dynamics is disturbed by the fact that the quasi-kinetic energy inherits the indefiniteness of the AdS 3 metric, giving a negative 'mass' to Ψ 1 and Ψ 2 . The Yang-Mills Lagrangian on the cylinder I × PSL(2, R) becomes and the Euler-Lagrange equations derived from (6.9) coincide with (6.6).

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where the number of minus signs in each triple must be even. The Ψ α =0 configuration corresponds to the vacuum solution F=0, while the other critical points yield genuine non-Abelian Yang-Mills solutions.

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Yang-Mills solutions with finite action. Now we consider the metric (5.16) on AdS 4 . Thanks to the conformal invariance of the Yang-Mills equations in four dimensions, it suffices to study Yang-Mills theory on S 1 ×S 3 + with the metric (5.18) including the standard metric (B.2) restricted to the upper hemisphere S 3 + . Since the one-forms e a in (5.17) obey (2.4), we can literally copy the ansatz (3.2) for I × S 3 to our space and restricting e a to the upper hemisphere S 3 + . The bounce solution (3.18) does not qualify. However, it is the limiting case of a continuum of periodic solutions given by a Jacobi elliptic function, with α(k) = 2/(2−k 2 ) and 0 ≤ k < 1 . (6.20) This family interpolates between the bounce (3.18) for k → 1 (period → ∞) and the constant vacuum solution ψ = 1 for k = 0. For infinitesimally small values of k we have harmonic oscillations with a period of π, as can be gleaned from a harmonic approximation to (3.15). By continuity, shorter periods cannot be attained, but there should exist a 2πperiodic solution for a special value of k. Since dn[u; k] has a period of 2K(k), where K(k) is the complete elliptic integral of the first kind (see e.g. the appendix of [14] for a brief discussion of Jacobi functions), the periodicity condition is satisfied if [18,19] K(k) α(k) = π ⇒ k =k ≈ 0.9977 , (6.21) which is very near to the bounce, as can be seen in figure 4. We lift all these solutions from the cylinder (5.18) to AdS 4 with metric (5.16) written as for the colour-magnetic solution, whereS 3 + is the hemisphere with its metric conformally rescaled by R 2 cos 2 χ . On the configuration (6.23), the energy evaluates to We see that it is not only finite but, contrary to the de Sitter case, it does not depend on time. Hence, we get a static magnetic configuration. For the configuration (6.24) we use

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results about sphalerons on a circle [18,19] to calculate the energy and find which correctly interpolates between the values for the vacuum (k = 0) and the static magnetic solution (k → 1). For the admissible value of k, its value E(k) differs from (6.27) by a factor of about 1 − 5×10 −6 . Finally, the energy of the Abelian configuration (6.25) is where γ 2 is the moduli parameter. So, for all three Yang-Mills solutions the energy is finite and constant.
Action of the Yang-Mills solutions. As previously, the action functional on the field configurations (6.23)-(6.25) can be calculated either in the anti-de Sitter metric (5.16) or in the cylinder metric (5.18) on S 1 ×S 3 + , For the purely magnetic configuration (6.23) the action evaluates to Interestingly, it coincides with the value (3.31) for the analogous but non-static configuration on dS 4 . This is because they are identical but are lifted from different spaces which happen to have the same volume, vol(I×S 3 ) = π × 2π 2 = 2π × π 2 = vol(S 1 ×S 3 + ) . (6.32) The action of the configuration (6.24) is reduced to where ψ(t; k) is periodic and given in (6.20). This integral is finite and independent of t 0 but cannot be written down analytically. Its numerical value is about 41% of (6.31). Finally, the action functional on the Abelian solution (6.25) vanishes, because the integral of the electric and magnetic energy densities are finite and equal.

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Boundary values of the Yang-Mills solutions. Since anti-de Sitter space has a boundary, it is of interest to note the value our solutions take there. The infinite-action field components (6.13) and (6.15) as well as the finite-action fields (6.23b), (6.24b) and (6.25b) all carry the conformal factor cosh −2 z = cos 2 χ, which vanishes at the boundary z → ±∞ or χ = ± π 2 . Therefore, our solutions live in the subspace of gauge fields decaying to zero at the AdS 4 boundary.
7 Self-dual Yang-Mills fields on anti-de Sitter space AdS 4 Euclidean AdS 4 . Finally we discuss instantons in anti-de Sitter space AdS 4 . The Euclidean continuation of the AdS 3 metric (C.2) is obtained by substituting t = iτ , which turns the z=const slices to 3-dimensional hyperbolic spaces H 3 . The metric on AdS 4 transforms to a cosh-cone metric on the hyperbolic space H 4 . This form of metric on H 4 is not convenient for our study of instantons since the natural boundary of H 4 is the 3-sphere S 3 = ∂H 4 . However, there exist various other choices of coordinates and metrics on AdS 4 (see e.g. [20]), such as Choosing t = i(χ− π 2 ) in (7.1c) one obtains for H 4 a sinh-cone metric over S 3 , ds 2 = R 2 dρ 2 + sinh 2 ρ (dχ 2 + sin 2 χ dΩ 2 2 ) = R 2 (dρ 2 + sinh 2 ρ δ ab e a e b ) , which is convenient for analyzing gauge instantons on AdS 4 . 8 Here, e a are the left-invariant one-forms on S 3 satisfying (2.4) and discussed in detail in appendix B. We remark that, due to the range ρ > 0 the metric (7.2) describes only one sheet of the two-sheeted hyperboloid in R 4,1 as a complete model of Euclideanized AdS 4 . Furthermore, we must eventually fix some boundary conditions for the gauge fields, in order to investigate stability, for instance. Here, we take the attitude to postpone this discussion and first learn about classical solutions for any kind of boundary condition.
Cylinder metric. In stereographic coordinates x i , i = 1, . . . , 4, the H 4 metric reads which resembles the metric (4.5) on S 4 . The forms (7.3) and (7.2) are related by the coordinate transformation

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Further, the metric (7.2) is conformally equivalent to the metric (4.10) on the Euclidean cylinder, BPST-type quasi-instanton. The Yang-Mills self-duality equations (4.9) are valid on any four-manifold. For the metric (7.5) on H 4 they are reduced to the equations on the cylinder R × S 3 with the metric (4.10) and become the generalized Nahm equations (4.12) for three matrices X a . Therefore, we can copy the kink solution presented in section 4, where the I a are defined by (3.5) and T 0 is a real parameter. Thus we see that up to this moment the analysis of the self-dual Yang-Mills equations is the same on R 4 (as a metric-cone over S 3 ), on S 4 (as a sine-cone over S 3 ), or on H 4 (as a sinh-cone over S 3 ).
The differences appear only in the range of T and in the role of moduli parameter T 0 . First, for the cylinder R×S 3 we have T ∈ (−∞, ∞), yielding where g(χ, θ, φ) : S 3 → SU(2) is a smooth map of degree (winding number) one. Thus, A(T ) describes a transition from the trivial vacuum (sector of topological charge q=0) to a nontrivial vacuum (sector q=1). Second, for the sphere S 4 one takes T ∈ [−∞, ∞], corresponding to ϕ ∈ [0, π]. Hence, the self-dual solution again has topological charge q=1 and extends the one from R 4 to S 4 . In more detail, for the gauge field from (7.7) we get where e 4 := dT . It follows that This integral depends neither on the metric nor on T 0 and is the same for R 4 , R×S 3 and S 4 . Third, turning to hyperbolic space H 4 , we see that This means that our solution (7.7) and (7.9) is defined only on the half line R − and describes a transition i.e. F(T =−∞) = 0 to F(T =0) = 1 4 cosh 2 T 0 η a ij e i ∧e j I a , (7.12)

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connecting the trivial vacuum with an instanton section of size Λ = e T 0 R, as discussed in [22]. Its quasi-topological charge depends on the moduli parameter T 0 , ranging from q=0 for T 0 →∞ to q=1 for T 0 →−∞. For T 0 =0 the boundary configuration sits in the middle of the kink, and so the solution (7.7) and (7.9) on R − corresponds to a meron [23] (a singular non-self-dual Yang-Mills solution) which has topological charge q = 1 2 in agreement with (7.13). We also see that for self-dual configurations on R − ×S 3 (and hence on H 4 ) the action functional S = 4π 2 C(j) |q| decreases monotonically with T 0 .
Geometric quasi-instanton. When studing instantons on AdS 4 , one more possibility opens up. The simple flow equation (4.19) has, besides the kink in (7.7), also the singular solution ψ(T ) = coth(T −T 0 ) . (7.14) It must be discarded on R 4 and on S 4 due to the pole at T = T 0 . However, for T 0 > 0 there is no singularity on the domain [−∞, 0) ∋ T relevant for H 4 . Substituting (7.14) into (7.7), we obtain the self-dual solution With (4.8) and (4.22) we get and (7.15) coincides with the self-dual Yang-Mills configuration on H 4 naturally appearing in the geometric construction of [24], for example. 9 The topological charge of (7.15) comes out as where now T 0 > 0. Thus, just like (7.9), the solution (7.15) has finite action.
Instantons in AdS 4 . For a fuller picture we take a look at self-dual solutions on the universal cover AdS 4 . To this end we perform a Euclidean continuation of the metric (5.16) as proposed in [21],

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where T ∈ S 1 for AdS 4 but T ∈ R for AdS 4 . As before, S 3 + denotes the upper hemisphere, with a volume of π 2 . The conformal boundary of this metric corresponds to χ = π 2 and has the topology of a Euclidean space S 1 × S 2 or R × S 2 , respectively.
As before, the self-duality equations reduce to equations on the cylinder, but over S 3 + instead of S 3 . After taking the canonical ansatz (4.11) and specializing to (4.13) and (4.18) we again obtain the flow equation (4.19). However, this equation admits no periodic solution, so we do not find BPS configurations on Euclideanized AdS 4 in this way. On the other hand, the Euclildean version of the universal cover AdS 4 relaxes the periodicity requirement. Therefore, on this space we can take the canonical kink solution (7.7) defined for T ∈ R. Then the gauge field (7.9) has the unit topological charge (7.10). Thus, standard instantons are well defined on the Euclidean version of the universal covering AdS 4 of anti-de Sitter space. Non-self-dual Yang-Mills solutions on Euclideanized AdS 4 can nevertheless be found. The trusted ansatz (4.11), (4.13) and (4.18) reduces the full Yang-Mills equations to whose periodic solutions in terms of Jacobi elliptic functions are described e.g. in [13]. Substituting back into the ansatz yields non-self-dual finite-action Yang-Mills configurations, which describe a sequence of instanton-anti-instanton pairs.

Conclusions
We have established the existence of solitonic classical pure Yang-Mills configurations with finite energy and action in four-dimensional de Sitter and anti-de Sitter spaces. No Higgs fields are required. On de Sitter space dS 4 described as spatial S 3 slices over real time, our Yang-Mills solutions are spatially homogeneous and decay exponentially for early and late times. Replacing S 3 with AdS 3 yields infinite-action configurations on AdS 4 . However, on anti-de Sitter space AdS 4 parametrized as spatial H 3 slices over a temporal circle, we again constructed solutions having finite energy and action, which decay exponentially in the radial direction of the hyperbolic slices. 10 For the Euclideanized version of R 3,1 , dS 4 and AdS 4 , our method reproduces the known BPST instantons and lifts them from R 4 to S 4 and H 4 , respectively, where in the latter case we get only 'half' the instanton. Due to their finite action, the described gauge configurations should be relevant in a semiclassical analysis of the path integral for quantum Yang-Mills theory on dS 4 or AdS 4 . Their existence indicates that the Yang-Mills vacuum structure may depend on the cosmological constant, and the question of their stability calls for a computation of the (one-loop) effective action around these field configurations. One might hope to employ the (anti-)de Sitter radius R as a regulator towards quantum Yang-Mills theory on Minkowski space.
The most symmetric solution has an elementary geometric dependence on de Sitter time or on anti-de Sitter radial distance, and its action in both cases takes the minimal JHEP11(2017)017 value of −3π 3 (for the SU(2) adjoint representation with normalization (3.5)), independent the (anti-)de Sitter radius. We conjecture this to be a lower bound for Yang-Mills solutions on these backgrounds. It will be important to investigate the stability of our configurations for certain boundary conditions. Our solutions derive from three simplifying ansätze. First, we restricted the gauge potential to an su(2) subalgebra and made an SU(2)-equivariant ansatz, which turns the Yang-Mills equations into ordinary coupled differential equations for three matrices. Second, we took these matrices to be proportional to the SU(2) generators, which produces a 3dimensional Newtonian dynamical system with tetrahedral (S 4 ) symmetry. Third, we focus on stable submanifolds in the parameter space, which enables us to find analytic solutions.
At each step, generalizations are possible. First, one may admit a larger gauge group and a more general ansatz for the matrices, which will lead to quiver gauge theories. Second, it is tempting to analyze the matrix dynamics directly, for the potential and superpotential respectively. And third, for a good understanding of the analog Newtonian system one should investigate also numerical solutions for its full 3-dimensional dynamics. We hope to address these issues in the near future.

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where the angles 0 ≤ θ ≤ π and 0 ≤ φ, χ ≤ 2π differ from those used above (but are denoted the same). For (B.5) substituted in (2.5) one obtains Correspondingly, the induced metric on the unit 3-sphere reads This metric is adapted to the Hopf fibration with the one-monopole connection 11 entering the metric (B.7).

D Yang-Mills solutions on dS 4 in various coordinates
We have constructed pure SU(2) Yang-Mills solutions in dS 4 parametrized by classical double-well trajectories ψ(t). It is remarkable that their action is finite. Their scale is set by the inverse de Sitter radius R −1 . Here we display these solutions in different coordinates on de Sitter space.

D.1 Yang-Mills configuration in closed slicing
The solutions found in [9] for the closed slicing and described in more detail in section 3 depend on a suitable function ψ(τ ) and have the form (D.8) From these expressions, it is straightforward to write down the components of A and F on the 3-sphere, namely A τ = 0 and The corresponding electric and magnetic field components are then read off as and we see that the geometry factors precisely cancel for the magnetic components, hence (D.11) Inspecting the matrices we see that all components are completely regular. To view our fields on de Sitter space, we introduce an orthonormal basis on dS 4 , e 0 := dτ = R dτ andẽ a := R cosh τ e a , (D. 12) and expand A = A aẽ a and F = F 0aẽ 0 ∧ẽ a + 1 2 F bcẽ b ∧ẽ c (D.13) so that A a = R cosh τ A a , F bc = R 2 cosh 2 τ F bc , F τ a = R 2 cosh τ F 0a . (D.14) Therefore, the electric and magnetic field components in closed-slicing coordinates are (D.16) The de Sitter energy of the Yang-Mills configuration then turns out to be (ψ = cosh τ dψ dτ ) Eτ = In the adjoint representation, we arrive at (D.20) The expression for the field strength is straightforward but too lengthy to write down here.

D.3 Yang-Mills configuration in static slicing
With a common 2-sphere parametrization, the relation between the closed and the static slice is given by just two relations, e.g.
In order to evaluate the components of the gauge potential and the field strength in the static coordinates (x I ), we transform them from the closed coordinates (x i ) according to and re-express the arguments x i = x i (x I ), e.g.
sin χ ⇒ ∆ −1 sin α and cos χ ⇒ ∆ −1 cos α cosh t . (D.28) Since the S 2 coordinates θ and φ are common to both systems and we employ the A τ = 0 gauge, we remain with In these coordinates electric and magnetic field components are defined as Passing to the dimensional (tilded) coordinates multiplies these relations with a factor of The radial components expressed in terms of the static coordinates take a reasonably simple form, (D.33) It will be interesting to physically interpret the static field components, in particular for the limits α → 0 ('center' of the configuration) and α → 1 (cosmological horizon).